259
J. Fluid Mech. (1996), 001. 316, p p . 259-284
Copyright @ 1996 Cambridge University Press
Chaotic advection in three-dimensional unsteady
incompressible laminar flow
By J U L Y A N H. E. CARTWRIGHT1>2,M A R I O F E I N G O L D 3 ,
AND O R E S T E P I R O ~ > ~
'Departament de Fisica, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain;
email [email protected]; [email protected]
*Centre de Calcul i Informatitzacib, Universitat de les Illes Balears,
07071 Palma de Mallorca, Spain
3Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel;
email [email protected]
41nstitut Mediterrani d'Estudis AvanGats (CSIC-UIB), 07071 Palma de Mallorca, Spain
(Received 31 March 1995 and in revised form 29 August 1995)
We discuss chaotic advection in three-dimensional unsteady incompressible laminar
flow, and analyse in detail the most important novel advection phenomenon in these
flows: the global dispersion of passive scalars in flows with two slow and one fast
velocity components. We make a comprehensive study of the first model of an
experimentally realizable flow to exhibit this resonance-induced dispersion : biaxial
unsteady spherical Couette flow is a three-dimensional incompressible laminar flow
with periodic time dependence derived analytically from the Navier-Stokes equations
in the low-Reynolds-number limit.
1. Introduction
There is a hierarchy in the dynamics of incompressible laminar? fluid flows. A flow
can be one-, two-, or three-dimensional, and it may be steady (time independent) or
unsteady (time dependent). The character of a laminar flow varies greatly depending
on which category in this hierarchy it belongs to. Of course, a fluid possesses three
spatial dimensions, and in this trivial sense a fluid flow is always a three-dimensional
entity. However, in many instances the fluid velocity is zero in one or more coordinate
directions, and here one can speak of a one- or two-dimensional flow. Another possible
way for a one- or two-dimensional flow to arise - a generalization of the above is that one or more of the three velocity components may be independent of the
others. This category, which might be termed that of (2 1)-dimensionalflows, comes
between truly two-dimensional flows where the third velocity component is zero, and
truly three-dimensional flows where all three velocity components are coupled. An
example of (2 1)-dimensional flow is spherical Couette flow: the steady flow at low
Reynolds numbers between coaxially rotating concentric spheres, where there is both a
primary one-dimensional flow and a secondary two-dimensional flow mathematically
+
+
t We are using laminar to mean non-turbulent here - the (Eulerian) velocity at any given point
in space is fixed, periodic, or quasiperiodic in time - following the generally established usage. It
should be born in mind, however, that the laminae are not necessarily either topologically simple
or fixed in time (Aref et al. 1989).
260
J. H . E. Cartwright, M . Feingold and 0. Piro
completely independent of the primary flow, although physically induced by it.? In
this paper we take advantage of the spherical symmetry of the geometry of this system
to couple the primary and secondary flows. The coupling produces a truly threedimensional flow - biaxial unsteady spherical Couette flow - which we can model
analytically as a solution of the Navier-Stokes equations. The purpose of this paper is
to study in detail the dynamics of biaxial unsteady spherical Couette flow. We present
results showing that a novel and important phenomenon of global resonance-induced
particle dispersion takes place, and show that such resonance-induced dispersion is
generic in three-dimensional unsteady incompressible laminar flows with two slow
and one fast velocity components.
We are interested in particle paths (pathlines) in laminar flow. Idealized particles
that are small enough not to perturb the flow, but large enough not to engage in
Brownian motion (i.e. not to diffuse), and that move with the flow itself are known as
passive scalars. The same physics applies to scalar or vector quantities possessed by
a fluid, such as temperature, concentration of a second fluid, or magnetic field. These
quantities move with the flow - are passively advected - and so may be described by
the Lagrangian representation of the fluid; their dynamics is that of a fluid element
where ux, uy, and u, are the Cartesian components of the velocity field u(ux,uy,u,).
We shall be studying laminar flows where u is time periodic (unsteady) or time
independent (steady), that are incompressible, that is to say V * u = 0.
Before embarking on our exploration of three-dimensional unsteady flows, we
briefly recall the state of knowledge of advection in laminar flows at lower levels of
the flow hierarchy, starting with the simplest case. One-dimensional incompressible
laminar flows, such as Poiseuille flow or Couette flow, are encountered at the beginning
of fluid-dynamics textbooks, precisely because their dynamics, whether steady or
unsteady, is very simple: one dimensionality plus incompressibility means that the
fluid velocity must be constant at any time in the direction of flow. In two dimensions,
au,/ay = 0, which tells us that
incompressibility of the flow implies that &,/ax
there exists an exact differential d y such that
+
y is known as the stream function, and we can immediately recognize these equations
as forming a Hamiltonian dynamical system. The motion of a passive scalar in a twodimensional incompressible flow can then be identified with the Hamiltonian dynamics
of a point in phase space, with the stream function y as the Hamiltonian. Accordingly,
two-dimensional steady incompressible flows are, like one-degree-of-freedom timeindependent Hamiltonians, integrable. Two-dimensional unsteady incompressible
flows, on the other hand, have the dynamics of one-degree-of-freedom time-dependent
Hamiltonians, which are generically non-integrable. In the case where the time
dependence is periodic, these have stroboscopic maps - mappings from one period to
the next - which are area-preserving maps on the plane, and show the chaotic motion
typical of these maps, with chaotic trajectories bounded by the invariant KAM tori
predicted by the famous Kolmogorov-Arnold-Moser theorem. The appearance of
chaotic streamlines in the Lagrangian picture of fluid dynamics has been termed
t Another flow that falls into this category is the eccentric helical annular mixer model of Ottino
(1989), and Kusch & Ottino (1992); regular duct flows in general are of this type.
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
261
chaotic advection, or Lagrangian turbulence? and is important not least because it
greatly enhances transport in these flows compared with their steady counterparts.
Both numerical (Aref 1984; Aref & Balachandar 1986; Chien, Rising & Ottino 1986;
Khakhar, Rising & Ottino 1986; Chaiken et al. 1987) and experimental (Chaiken
et al. 1986; Solomon & Gollub 1988; Swanson & Ottino 1990) investigations of
chaotic advection in two-dimensional unsteady incompressible laminar flows have
been undertaken. Many of these investigations have used as a model the unsteady
Couette flow between two eccentric cylinders, often termed unsteady journal-bearing
flow, for which an analytical solution of the Navier-Stokes equations is available in
the low-Reynolds-number limit. Unsteady eccentric Couette flow has proved to be an
ideal experimental and numerical model for chaotic advection in two dimensions.
The steady three-dimensional incompressible flow problem is similar to the unsteady case in two dimensions, since the return map to any section transverse to the
flow is an area-preserving map on the plane. Moreover, a steady three-dimensional
incompressible flow can be reduced to a one-degree-of-freedom time-dependent
Hamiltonian system, except at points where the flow has a stagnation point (in
the terminology of dynamical systems, a jixed point) (Arnold 1978; Bajer 1994;
MacKay 1994). If there are no stagnation points and if the flow in one coordinate direction is never zero, this coordinate can be used as a fictitious time (Aref et al. 1989;
MacKay 1994). Arnold (1965) first addressed chaos in three-dimensional steady incompressible inviscid fluid flow. He proved that if the vorticity w = V x u is nowhere
parallel to the velocity u then the flow is integrable. At the same time he conjectured
that Beltrami flows where vorticity and velocity are everywhere parallel, w = Au,
would have streamlines of complex topology. Htnon (1966) took up this suggestion
and examined the A = 1 case, now known as ABC (Arnold-Beltrami-Childress)
flow. The ABC flow, an inviscid flow - a solution of the three-dimensional Euler
equation - was subsequently studied by Dombre et al. (1986), and has become the
archetypal mathematical model for the study of chaos in three-dimensional steady
incompressible laminar flow. Interesting chaotic Beltrami flows in a sphere - the
spherical analogues of ABC flow - were analysed by Zheligovsky (1993). Other mathematical models of chaotic advection in three-dimensional steady Stokes flows within
a spherical drop have been investigated by Bajer & Moffatt (1990), and Stone, Nadim
& Strogatz (1991). Three-dimensional steady flows that are experimentally realizable
have also been shown to exhibit chaotic advection. The partitioned-pipe mixer model
of Khakhar, Frajione & Ottino (1987) has been experimentally studied by Kusch &
Ottino (1992). Similarly, the twisted-pipe model of Jones, Thomas & Aref (1989) has
led to experimental investigations by Le Guer, Castelain & Peerhossaini (1996).
Three-dimensional unsteady incompressible laminar flow has an entirely different
dynamics. We survey in 92 the types of chaotic advection to expect in the threedimensional unsteady case using a classification of these unsteady volume-conserving
flows by the number of fast and slow components of the velocity field. Concentrating
in 93 on flows with two slow and one fast velocity components, in $3.1 we review
spherical Couette flow, in order that in $3.2we may go on to fulfil our aim in this paper,
to explore in detail a model we (Cartwright, Feingold & Piro 1994a, 1995) recently
introduced of an experimentally realizable flow that exhibits the most important
novel advection phenomenon in three-dimensional flows : the global resonance-induced
dispersion of passive scalars. Finally, in 94, we draw these different threads together
t We prefer the former terminology to the latter, since it is not clear what relationship there is
between chaotic advection and turbulence.
262
J. H. E. Cartwright, M . Feingold and 0. Piro
and discuss the possibilities for experiments based on this and other unsteady threedimensional flows.
2. Classifying three-dimensional unsteady flows with Liouvillian maps
A classical technique, due to Poincare, for the study of a system of ordinary
differential equations describing a dynamical system such as a fluid flow, is the
reduction of the dimension of the system achieved by sectioning the flow in time
or space to produce a map. The dynamical behaviour of this map can then be
investigated much more easily than that of the original system, in the knowledge that
its limit sets are simply related to the limit sets of the flow (Arrowsmith & Place
1990). The technical details of this procedure differ depending on whether the flow
is autonomous (steady) or non-autonomous (unsteady) (Parker & Chua 1989), and
the terminology is not completely fixed: here we take a Poincare section to mean
any such section, and a Poincare map to be the dynamics on the section. We talk
of a stroboscopic map when we refer to the Poincare map of a non-autonomous
system obtained by sampling the flow stroboscopically once per period. The Poincare
map of an autonomous system obtained by taking the intersections of the trajectory
(pathline) with a fixed plane we refer to as a return map.
Three-dimensional unsteady incompressible flows with periodic time dependence
produce stroboscopic maps in three dimensions that preserve volume. These threedimensional volume-preserving maps do not correspond to Hamiltonian systems,
which have phase spaces of even dimensionality: Hamiltonians with two or three
degrees of freedom produce volume-preserving stroboscopic maps of two or four
dimensions respectively. Following Liouville’s theorem, we have named volumepreserving maps of three dimensions, which dimensionally interpolate between the
two cases above, Liouvillian maps (Feingold, Kadanoff & Piro 1987, 1988a,b, 1989;
Piro & Feingold 1988; Cartwright, Feingold & Piro 1994b).
To study Liouvillian maps one can employ ideas from KAM theory and imagine
perturbations of integrable action-angle models where the actions are fixed and the
angles change at each iteration. A perturbation away from the integrable case then
causes the actions to vary slowly. For two-dimensional area-preserving maps, or for
any maps derived from Hamiltonian dynamical systems, the underlying symplectic
structure provides the action-angle variables on which to build. Here with Liouvillian
maps such symplectic structure is absent, so we must define our variables with respect
to something else. Thus we use invariant tori to define the action-angle variables.
We can consider integrable, Liouvillian maps that possess some variables, actions I ,
that do not change from iteration to iteration, whereas others, angles 8, vary linearly
as the map is iteratedf:
+
z’ = I ,
el = e
w(z).
(2.1)
We use the notation LE to represent an integrable Liouvillian map which possesses
a set of action-angle variables Z E Rk,8 E T3-k.A k-action integrable Liouvillian
map LE has a family of ( 3 - k)-dimensional invariant tori, each of which is traversed
by the 3 - k angle variables that rotate at each iteration by an amount given by the
corresponding component of the ( 3 - k)-dimensional vector o. In other words, L:
t Notice that at this stage it is not important whether the frequency vector is made a function
of the old actions w(Z) or of the iterated actions w(Z’). This will become important when we wish
to perturb this map whilst retaining its Liouvillian nature: we construct the perturbation to make
it automatically volume preserving.
Chaotic advection in three-dimensional unsteady incompressible laminar flow 263
maps consist of uniform translations on (3 - k)-tori embedded in T3.We thus have
four possibilities for these integrable maps :
L:. Zero actions; three angles. The frequency vector o is constant, and hence we
have a uniform rotation on T3.
Li. One action; two angles. The motion takes place on two-tori defined by constant
I , and a point on one of these tori is given by the two angles 81 and 192. Thus the
motion on the tori is a uniform rotation with a frequency depending on the value of
the action.
Li. Two actions; one angle. The two actions parametrize a family of invariant
circles (i.e. one-tori), and the angle variable rotates with a frequency 0 ( 1 1 , 1 2 ) .
L:. Three actions; zero angles. Each point of space is parametrized by (11912,13)
and is hence invariant; the map is the identity.
We consider now the effects of non-integrable perturbations on L: maps. The
perturbations we wish to consider will act to couple the hitherto independent actions
and angles whilst retaining the Liouvillian property of the map. Thyagaraja & Haas
(1985) have worked out the most general form that such perturbations can take,
which in general leads to an implicit map. Below we shall discuss only the subset
of perturbations that leave the map in explicit form. We believe that these explicit
Liouvillian maps capture the dynamics of Liouvillian maps in general, whilst being
easier to work with than implicit maps.
We deal first with the two extreme cases k = 0 and k = 3. Results of Ruelle
& Takens (1971) and Newhouse, Ruelle & Takens (1978) about perturbing quasiperiodic flows on a three-torus indicate that small perturbations of zero-action L:
maps will generically lead to completely chaotic behaviourt. A small non-integrable
perturbation added to three-action L! maps is a small perturbation
of the identity. In the E + 0 limit we have a set of three autonomous ordinary
differential equations, which takes us back to the case of three-dimensional steady
flows. In other words, the perturbation represents the Euler map of the flow Z = F ( Z ) ;
the map (2.2) is derived from this flow by applying the forwards and backwards
Euler numerical integration methods to discretize it. The dynamics of Euler maps are
closely related to the flows from which they are derived (Cartwright & Piro 1992).
For one-action Li maps, it turns out that most of the invariant two-tori are
preserved in a KAM-like manner under small non-integrable volume-preserving perturbations. However, the resonant tori satisfying the equation
+
m o l ( I ) nw2(I) = 2nk,
(2.3)
where k, m, and n are integers, are destroyed, and in their place appear a finite
number of invariant circles, half of which are stable and the other half unstable, in
the same way as invariant circles are destroyed and fixed points appear in planar
area-preserving maps following the Poincarb-Birkhoff theorem. Further invariant
tori surround the stable invariant circles, whilst chaos appears around the unstable
ones. The preserved invariant tori act as boundaries for the chaotic layers, so a
single trajectory cannot cover the whole phase space until the perturbation is large
enough to destroy the final invariant torus. The whole situation is like that governed
by the KAM theorem in planar area-preserving maps, except that the structures
t However, the effect of restricting the perturbations so that they retain volume conservation has
not, as far as we know, been investigated.
264
J. H . E. Cartwright, M . Feingold and 0. Piro
have one additional dimension: that is the invariant tori are two- rather than onedimensional, the island chains interspersed amongst them contain invariant circles,
not fixed points, and the stable (elliptic) and unstable (hyperbolic) invariant circles
are centres for further invariant tori and for chaos respectively (Feingold et al. 1988b;
Cartwright et al. 199427). The existence of a KAM-like theorem in this one-action case
has been proven by Cheng & Sun (1990), although as Mezib & Wiggins (1994) point
out, in three dimensions we have no criterion similar to the irrationality of rotation
numbers in two dimensions that enables one to predict in which order the remaining
invariant tori are destroyed as the non-integrable perturbation is increased.
With two-action Li maps, on the other hand, small non-integrable volumepreserving perturbations cause behaviour rather different to that detailed above.
The integrable case turns out to be a singular limit: as soon as a non-integrable
perturbation is switched on, the two-parameter family of invariant circles coalesce
into invariant two-tori. We can see why this should be so using an argument based
on an adiabatic approximation. Consider the perturbed two-action map
Suppose that co is irrational, then when E is small, the angle 0 covers uniformly the
entire interval (0,27c) before the actions Z change significantly. The variation of Z is
thus sensitive only to the average of F over 8. Therefore the actions now iterate as
z;
= I1
+ &F1(12),
z;
= I2
+EF2(Zi),
(2.5)
where the bar represents the &average. Thus the dynamics of the actions Z decouples
from that of the angle 6’ for non-resonant a. We are left with an area-preserving
map, which leads in the limit E + 0 to the Hamiltonian system
aH
i, = Fl(iz)= -,312
i, = F2(i1)= --.aH
a11
We obtain the Hamiltonian
rIz
rli
In other words, in the E + 0 limit, the action variables evolve slowly along the level
curves of (2.7). Including the fast motion in the &direction, we infer that the originally
invariant circles parallel to the &axis coalesce into invariant tori Cg defined by the
condition H(Zl,I2) = p, the motion on which is fast in one direction and slow in the
other. Resonances occur when the condition imposed above that co(Z) be irrational is
not satisfied and we have
nw(Zl,I2) = 27ck,
(2.8)
where k and n are integers. The resonant condition defines a set of sheets that
typically intersect a continuous set of invariant tori, which break down locally at
the intersections. The remainder of the invariant torus survives, to a given order in
the perturbation expansion. Invariant tori are thus connected through resonances,
meaning that, in contrast with the one-action case, a single trajectory can cover the
whole of phase space (Piro & Feingold 1988; Feingold et al. 1989; Cartwright et
al. 1994b). We term this characteristic behaviour of two-action Liouvillian maps
resonance-induced dispersion.
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
265
3. Resonance-induced dispersion in two-action flows
As we have seen, two-action Liouvillian maps have particularly interesting behaviour, since there are no barriers to motion and a form of global dispersion is
present. To investigate this resonance-induced dispersion further in a fluid dynamical
context, in this section we construct a model of a realistic flow showing two-action
properties.
3.1. Spherical Couette Jlow
Spherical Couette flow - steady flow at low Reynolds numbers between coaxiallyrotating concentric spheres - is a natural generalization of the Couette flow between
two rotating cylinders. In contrast with the latter, however, it possesses both a
primary and a secondary flow, and it is dependent on the Reynolds number. References to experimental and numerical studies of spherical Couette flow are given by
Koschmieder (1993).
Consider concentric spheres with fixed radii R1 and R2, and constant angular
velocities about a common axis through their centres b, and 02, as pictured in figure
1. In the spherical polar coordinate system of figure 1, with the radial coordinate
rescaled as r = f / R 2 , the steady flow of a viscous, incompressible fluid contained
between the two spheres is assumed to be independent of the longitude 6. We
may then write the Navier-Stokes equations in terms of a stream function y in the
meridian plane, and a longitudinal velocity function Q
1 4
-Dy,=--Re
2Q
(Qrrcos 8 - Qesin 0)
r3 sin2e
+
2D2y
(y,r cos 8 - Ye sin 0)
r3 sin28
~
where
(3.3)
and the subscripts associated with y and Q denote differentiation.
The Reynolds number is defined as Re = bcRZ/v, where R2 is a characteristic
length given by the radius of the outer sphere, bcis a characteristic angular velocity,
and v is the kinematic viscosity. The characteristic angular velocity may be taken as
depending on which is the
that of the outer sphere b2, or that of the inner sphere
dominant influence on the flow in the case being considered. The boundary conditions
are firstly that y = y r = 0 on both spheres, when r = q and r = 1. Secondly, with
b as the physical angular velocity, and choosing 02 as the characteristic angular
velocity, the dimensionless angular velocity o = d / b 2= Q / r 2 sin28 should be o = p
on the inner sphere and co = 1 on the outer. Hence the other two parameters, apart
from the Reynolds number, are the radius ratio y = Rl/R2, and the angular velocity
ratio p = b1/b2.
The above problem can be solved analytically by a perturbation method in the
Reynolds number. Haberman (1962) gave the solution to first order in the Reynolds
number. Subsequently Munson & Joseph (1971) derived general results through to
seventh order in the Reynolds number. Here we need only the first-order approxima-
a,,
266
J. H . E. Cartwright, M . Feingold and 0. Piro
I
FIGURE
1. Two concentric spheres have radii R1 and R2, and constant angular velocities bl and b2
about a common axis through their centres. The space between themjs filled with an incompressible
viscous fluid. We term the ratio R1 to R2, q , and the ratio s21 to 0 2 , p. The radial coordinate is
rescaled as r = P/R* in order that it be dimensionless.
tion valid in the low-Reynolds-number limit R e 5 1 (with the higher-order solutions
one can obtain good approximations for Reynolds numbers in the hundreds). The
solution to first order in the Reynolds number is
sz = (alr2 +
where
3
-
sin2e,
Chaotic advection in three-dimensional unsteady incompressible laminar flow 267
and
(
A1 = Ey4 (2y5 - y4 - 16y3- 20y2 - 8y - 2) p
+ 2y5+ gy4 + 2or3 + 1 6 ~ 2+ - 2),
A2 = -Ey2 (( 10y7+ 25y6 + 40y5 + 57y4 + 48y3 + 20y2 + 8y + 2) p
- 2y7 - 8y6 - 20y5 - 48y4 - 57y3- 40y2 - 25y - lo),
A3 = -E
((6y6+ 24y5 + 35y4+ 32y3+ 15y2- 2y - 5 ) py2
+ 5y6 + 2y5 - 15y4- 32y3- 35y2- 24y - 61,
A4 = E ( (2y4 + 8y3 + 5y2 - 2y - 3) py2 + 3y4 + 2y3 - 5y2 - 8y - 2),
(3.7)
(3.8)
(3.9)
(3.10)
with
-- - 4(4@ + 16y5+ 40y4 + 55y3 + 40y21 +h 316y + 4)(y
(P -
I -
-
+ + 1)2'
1)2(y2 y
(3.11)
The dimensionless flow velocities are given by
(3.12)
(3.13)
(3.14)
so that
(3.15)
(3.16)
d = a l + - pa2
.
(3.17)
The solution shows a primary flow SZ about the axis of rotation, together with a
secondary flow given by y in the meridian plane. In general, one of the two spheres
dominates the flow pattern, generating a pair of vortices, one on each side of the
equator, whose sense of rotation depends on the angular velocity ratio p. The spheres
may corotate or counter-rotate, and in the latter case ( p negative) there is a part of the
(y, p ) parameter space where neither sphere is dominant and we observe two counterrotating vortices in each hemisphere (Yavorskaya et al. 1980), giving a total of four
vortices. Figure 2 shows examples of both the two-vortex and the four-vortex patterns.
Since all three velocity components depend only on r and 8, spherical Couette flow
is in the category of flows we named (2 1)-dimensional in $1. The secondary flow
stream function y is a one-degree-of-freedom time-independent Hamiltonian of the
type (1.2), so the steady flow is identifiable as an integrable Hamiltonian system.
Now consider the flow in terms of the categorization we discussed in $2. Inserting
typical values for the magnitudes of the variables and parameters (by which we mean
+
268
J. H. E. Cartwright, M . Feingold and 0. Piro
4
I1
2
0
-2
-4
-6
I11
-8
-10
0
0.2
0.4
0.6
0.8
1.0
7
FIGURE2. The regions of parameter space for which the different secondary flow patterns between
the spheres exist, and typical examples of the two-vortex and four-vortex flow are represented as
cutaway pictures showing the spheres together with some level surfaces of the vortices. When p = 1
there is rigid body rotation and no secondary flow. Either side of this line two vortices are present.
In region I the outer sphere is the dominant influence on the flow, and fluid is transported from the
poles to the equator near the outer sphere, and back near the inner sphere. In the two disjoint parts
of region I1 the inner sphere is the dominant influence, and fluid is transported in the opposite
direction, from the equator to the poles near the outer sphere, and back near the inner sphere. In
between, in the linguiform region 111, neither sphere is the dominant influence and there are four
vortices. These rotate such that fluid is transported to the equator near the inner and outer spheres,
and to the poles at the separatrix between the two vortices of each hemisphere.
typical of those we use throughout the rest of this paper) into (3.15)-(3.17) shows
that at the values of the Reynolds number Re < 1 for which (3.15)-(3.17) hold, the
one-dimensional primary flow i2 governed by q5 is hundreds to thousands of times
faster than the two-dimensional secondary flow y governed by r and 8. Thus we have
two slow action variables r and 0, and one fast angle variable 4 ; the flow falls into
the two-action category.
Chaotic advection in three-dimensional unsteady incompressible laminar jlow 269
-0
\
1.o
FIGURE3. A trajectory in biaxial unsteady spherical Couette flow illustrates the piecewise (mathematically speaking, Co)nature of the solution. The trajectory covers thirty periods. The radius
ratio is '1 = 0.5, the angular velocity ratios are p1 = 0 and p2 = -3, and the semiperiods are
T I = T2 = T / 2 = 2. The Reynolds number is Re = 0.1 and the axis separation is LY = lo". The
initial conditions are r = 0.64, 0 = 60", and q5 = 30".
3.2. Biaxial unsteady spherical Couette jlow
The existence of a secondary flow in spherical Couette flow is crucial for our purposes,
since it enables us to construct a fully three-dimensional flow simply by using two
different axes of rotation for the spheres. Rotating periodically about first one axis,
then the other acts to couple the longitude to the latitude and the radial coordinate,
giving a three-dimensional flow that is also time dependent, figure 3. We call the time
for which we rotate about the first axis, the first semiperiod, TI, and that about the
second axis, the second semiperiod, T2, making a total forcing period of T = TI T2.
For simplicity we shall set T I = T2 = T/2, and we shall vary T as necessary. In the
following, all our numerical results will be obtained by the numerical integration of
the Navier-Stokes equations for spherical Couette flow given in (3.15)-(3.17).
Biaxial unsteady spherical Couette flow is then a piecewise combination of the
+
270
J. H. E. Cartwright, M . Feingold and 0. Piro
flow solutions about each axis. This piecewise combination of solutions is the same
technique used to obtain the unsteady journal-bearing flow extensively investigated
as a model for chaotic advection in two-dimensional unsteady flows. However, an
important conceptual difference between spherical Couette flow and journal-bearing
flow is that in the latter problem the solution depends linearly on the rotation rates of
the inner and outer cylinders. Thus in journal-bearing flow, particle paths depend only
on the boundary displacements. As long as the cylinders are set in motion alternately,
the details of the modulation of the rotation rate are not important; provided the
angular displacements of the boundaries are the same, so will be the path of a passive
scalar. This is not true of spherical Couette flow, since the stream function is not a
linear function of the rotation rate. Thus in the spheres case we have to decide the
way in which to modulate the rotation. The two extremes are the discontinuous limit,
and the quasi-static limit. In the discontinuous limit we change the rotation rate as
discontinuously as possible, with instantaneous startup and shutoff of the rotation,
as with a square wave. We then model the problem as a piecewise-linear combination
of the two separate steady spherical Couette flow solutions. This is a reasonable
approximation if the fluid inertia is negligible, so that the fluid stops moving as soon
as the sphere stops rotating, which is why we require low-Reynolds-number flow. In
the quasi-static limit, on the other hand, we change the rotation rate as slowly as
possible, with smooth modulation of the rotation in time, as with a sine wave. The
quasi-static approximation then consists of considering the resulting stream function
to be given instantaneously by the steady stream function for the rotation rate at
that instant. We have chosen to use the discontinuous approximation to model the
unsteady flow here. We shall use radius ratio q = 1/2 throughout this paper, and we
can elect to use different angular velocity ratios pl and p2 in each semiperiod to alter
the character of the flow solution.
The alternative to the above choices for modulating the rotation rates of the spheres
would be to model the full developing flow, the so-called spin-up problem discussed
by Pearson (1967). This makes the model far more complex, and we feel justified in
ignoring the complication since the discrepancy is small at low Reynolds numbers. In
the case of either unsteady journal-bearing flow or biaxial unsteady spherical Couette
flow, one is neglecting transient effects such as the diffusion of momentum. This
neglect requires that the acceleration terms in the Navier-Stokes equation
(3.18)
be negligible, where Re = V L / v is the Reynolds number as before, and Sr = f L / V is
the Strouhal number, with L a characteristic length, V a characteristic velocity, and
f a characteristic frequency; v is the kinematic viscosity. To make the acceleration
terms negligible, Re and Sr must be small. This last condition is violated by the
discontinuous modulation of the rotation rate in time; at the discontinuities, au/at
will be infinite. However, we argue that the effects of this violation will be slight at
low Reynolds numbers (many of the studies of journal-bearing flow have been made
with discontinuous changes in the cylinder rotation rates), and to place us squarely
in the low-Reynolds-number regime we set Re = 0.1 throughout this paper. Because
the dynamical phenomena we observe in this system are structurally stable, that is to
say robust against small changes in the system, the small extra displacement suffered
by a fluid particle not responding instantaneously when the flow stops and restarts
will not lead to a qualitative change in its long-term behaviour. This robustness
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
27 1
is also a prerequisite for being able to observe these phenomena in a real fluid
experiment.
To explore the dynamics of a three-dimensional unsteady flow, we may plot a
stroboscopic map, which gives us a three-dimensional volume-preserving map like
the Liouvillian maps of $2, but a three-dimensional view of a Liouvillian map often
fails to give much information. In order to reveal the information contained within
a three-dimensional map, we need to open it up to view. There are various methods
to achieve this; we shall use two in this paper. The first is to slice the map open
by taking a section through it. We select a thin slice and record the iterates that lie
within it. There is a trade off between the thickness of the slice and the quality of the
information that is obtained: a thicker slice has more points lying within it, but is
fuzzier because of its three-dimensionality, a thinner slice gives a sharper image but
requires longer time integration of the system to have the same number of points in
the slice. In the spheres case we use a slice between -0.01 < y < 0.01. The second
technique we use is to project the map down onto a plane. Unlike the slice technique,
which will always produce useful information, this technique is only useful if the
Liouvillian map is near to being integrable. If we are in possession of the action and
angle variables, projecting down onto an action-angle plane serves to fold up all the
information along the other action or angle variable. In the spheres case, with small
axis separation, this can be achieved by plotting a projection of the stroboscopic
map onto the action plane. We do this by taking the points of the stroboscopic
map expressed in action-angle variables and forgetting about the angle coordinate :
plotting the points only in the actions. One can think of the operation geometrically
as concertinaing the spheres about the angle variable into one plane.
Apart from the period T , the other new parameter in biaxial unsteady spherical
Couette flow is the angle a between the two axes. We have chosen to have the two axes
lying in the (x,z)-plane of our coordinate system (see figure 3), since this simplifies the
algebra whilst not affecting the dynamics. If a = 0, the spheres rotate about a single
axis, and we have a (2 1)-dimensional flow that may be steady, if p1 = p2 - which
corresponds to the spherical Couette flow described in $3.1 - or unsteady if p1 # p.2.
Whilst in the former case the flow is integrable, in unsteady spherical Couette flow
the spheres rotate about a single axis with angular velocities modulated with period
T , and the secondary flow stream function +
I may be identified with a one-degree-offreedom time-dependent Hamiltonian, which will generically exhibit chaos. Although
the secondary flow in ( r , 6) is decoupled from the primary flow in 4, it provides input
to the primary flow equation (3.17), which will be a chaotic input when the secondary
flow is chaotic?. Numerically integrating the flow with a = 0 for angular velocity
ratios p1 and p2, where p1 # p2, we find the Hamiltonian chaos we expected, as we
show in figure 4. We find that a significant measure of Hamiltonian chaos appears
in the flow only for large values of T : for low-Reynolds-number flow Re = 0.1, the
period T needs to be approximately lo4 before Hamiltonian chaos becomes apparent
in numerical simulations. The increasing amount of Hamiltonian chaos with longer
periodicities has also been noted in the investigations of journal-bearing flow.
If we introduce a small coupling between (r,O) and 4, by using two rotation
+
+
7 Thus in order to fully understand transport in ( 2 1)-dimensional flows like spherical Couette
flow, it is not enough to consider the Hamiltonian dynamics of the secondary flow alone. Nevertheless, it is possible to determine many properties of the flow from the secondary flow dynamics, and
since the secondary flow equations (3.15) and (3.16) have the Reynolds number Re as an overall
multiplying factor, if one is only interested in the secondary flow dynamics, the product Re T can
be treated as a single parameter.
272
J. H. E. Cartwright, M . Feingold and 0. Piro
1.c
..
...
,
,
I
. ...,.
1.c
I
0.5
0.5
2 0
a
-0.5
-0.5
-1.0
0
1.0
0.5
X
0
0.5
1.o
X
FIGURE
4. Projections of stroboscopic maps onto the plane y = r sin 6 sin 4 = 0 show Hamiltonian
chaos in the flow when CI = 0. Both projections show iterates resulting from several different initial
conditions. The parameter values in both cases are y~ = 0.5, Re = 0.1 and T = 12000. (a) p1 = 0,
p2 = -8, and the flow is a chaotic perturbation of two-vortex flow. ( b ) p1 = 0, p2 = -3, and the
flow is a chaotic perturbation of four-vortex flow.
axes with a small angle tl between them, we can simultaneously introduce both
time dependence and fully fledged three-dimensionality to the problem. In order to
analyse this, we take spherical Couette flow (3.15)-(3.17) and derive a Liouvillian
map describing the strobed behaviour of biaxial unsteady spherical Couette flow in
the limits of low Reynolds numbers and small axis separation. We have no reason
to favour one of the axes over the other, so it is natural to use a coordinate system
located in the a/2-rotated reference frame (r, e",6)oriented midway between the two
axes, that collapses back to ( r, 8 , 4 ) when tl = 0. In this frame the map can be written
r ' = r + $ R e ( f l ( r ) + f 2 ( r ) ) ( 3 c o s 2 e " - 1) T
-:Re
e"'= e"+
-$
(fl(r)cos6-f2(r) (2cosl -cos$)) sine"cose"Ta+ 0 ( a 2 ) ,
(3.19)
tRe(gl(r)+g2(r))sine"cos8T
+
(2cosl - c o s ~ - c o s ~ iRefl(r)hi(r) (3cos2e"- 1) sin5 T 2
-$Re (gl(r)cos
6- g2(r)(2cos 5 - cos 6))(2c0s2e" - 1) T ) tl + 0 ( a 2 ) ,
(3.20)
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
8' = 6+ i (hl(r) + h2(r))T + $Refl(r)hi(r) (3 cos2e" - 1) T 2
273
6- cosec 5: + ;Re ((gl(r)+ g2(r)) cosec 5: - 2gl(r)cosec [
+ifI(r)hi(r) ( ( 3 cos2 e" - 1) cosec 5 cot 5: - 6 sin2
6)T ) T ) cot $ a
+;
(2 cosec 5 - cosec
~"COS
(3.21)
+0(a2),
where h;(r)
= dhz(r)/dr,
5 = 6+ ihl(r)T,
and
5: = 6+
(hl(r)
+ h2(r)) T,
(3.22)
(3.23)
+3A3jr2
+ 5A,r
(3.24)
(3.25)
and j is 1 or 2; aij and A , are ai (3.6) and Ai (3.7)-(3.10), respectively, evaluated
with p = p j . The terms of order a are those through which r and e" couple to
and perturb the (2 1)-dimensional flow into being fully three-dimensional. Note
that when p1 = p2, so that fl(r) = f 2 ( r ) , gl(r) = g2(r), and h l ( r ) = h2(r), the two
semiperiods are identical and there is cancellation within the the order-a terms which
results in the coupling being weaker than otherwise. We emphasize that we shall be
using the map (3.19)-(3.21) and results derived from it to analyse the behaviour of
biaxial unsteady spherical Couette flow, but that we continue to integrate numerically
the flow itself taking the full Navier-Stokes solution of (3.15)-(3.17).
When Re and a are small, The map (3.19)-(3.21) has the same form as the
perturbed two-action Liouvillian map (2.4), with r and e" as the actions and as the
angle variable. As we showed in $2, a map of the action plane (r, 8) can be constructed
by averaging (3.19)-(3.20) over the angle giving
6
+
6,
r' = r + ;Re (fl(r) +
f2(r))
e' = e"+
6
(3 cos2 e" - 1) T ,
iRe(gl(r)+g2(r))sin8cose"T.
(3.26)
(3.27)
Since, as we mentioned above, R e T can be treated as a single parameter for the
purposes of the secondary flow, taking either of the limits Re + 0 or T t 0 of the
Euler map (3.26)-(3.27) has the same effect, and the actions evolve according to the
stream function
H
=
i r 2 (fl(r)
+ f 2 ( r ) )sin2Ocos e",
(3.28)
(3.29)
The invariant tori in the small-axis-separation limit are then the level surfaces of
(3.29). Note that (3.29) reduces to (3.4), up to a constant, when pl = p2. The resonant
surfaces in the small-axis-separation limit may be obtained as in (2.8) by considering
the values of the action variables for which the above averaging procedure fails, when
the angle variable changes by a rational fraction of 27c each period. We consider
274
J. H. E. Cartwright, M . Feingold and 0. Piro
1
1
z o
0
-1
-1
1
0
1
,
0
-1
A
1
A
FIGURE
5. Primary resonances (3.31) - shown dashed - and adiabatic invariant surfaces (3.29) in
biaxial spherical Couette flow are shown for small axis separation in the low-Reynolds-number
regime. ( a ) y = 0.5,p1 = 0, p2 = -8, and T = 4; a two-vortex flow. The resonances in the spheres
are shown where, from the inner sphere outwards, k / n is -2/1, - l / l , and 0/1. ( b ) = 0.5, p1 = 0,
p 2 = -3, and T = 20; a four-vortex flow. The increase in the axis-switching period T from ( a )
means there are more resonances in this the picture: those where k / n is -4/1, -3/1, -2/1, -l/l,
011, 111, 211, and 3/1 are shown.
(3.21) in the limits Re
and
-+
0 and a --+ 0, so that the terms of order a and Re disappear
- - 271k
#-$=-=-(
h ( r ) h20.1) T .
(3.30)
n
The resonances are then the solutions of (3.30) where k and n are integers. These
occur when
;
+
(3.31)
which are shells of constant r, whose radius depends on the mean value of p1 and p2,
= (pl
p 2 ) / 2 , as well as on y, T and the order k / n of the resonance. For example,
in figure 5 ( a ) we show the distribution of some of the resonances in a two-vortex
flow with y = 1/2, p1 = 0, p2 = -8, and T = 4.The terms of order a and R e in (3.21)
constitute negligible perturbations of these resonance shells when the axis separation
and the Reynolds number are small. The resonances are dense throughout space,
but they have a hierarchical structure in which their relative strengths are governed
by the Fourier expansion of (3.19)-(3.21), such that in general the resonances with
smaller denominators n are stronger: the primary resonances where n = 1 have the
largest influence, and the effect of the secondary and higher-order resonances n 2 2
is far smaller. One can tune the resonances using ji and T so that more or fewer
primary resonances be inside or outside the region y < r < 1 between the spheres. In
figure 5 ( b ) we show an example with y = 1/2, p1 = 0, p2 = -3, and T = 20 of a
four-vortex flow with more primary resonances than figure 5 (a).
In the projection of the stroboscopic map of the flow in figure 6 (a),we can see that
the trajectory is, as predicted, lying on the invariant tori of (3.29), except for intervals
where it jumps from one invariant torus to another. Also shown dashed in figure 6 ( a )
is the principal resonant surface, and it is apparent that the resonances are the means
ji
+
Chaotic advection in three-dimensional unsteady incompressible laminar flow 275
0.7
0.8
Z
0.4
27t
0.7
0.4
X
0.7
0.7
0.4
xl2n
FIGURE6. (a) A projection of a stroboscopic map onto the plane y = r s i n J s i n 6 = 0 illustrates
the jump behaviour typical of resonance-induced dispersion in two-action flows. The parameter
values are as in figure 5 (a), with Re = 0.1 and a = 0.1”. These parameter values are such as
to give a two-vortex flow, and to put the primary 0/1 resonance in the middle of the region
between the spheres, whilst leaving other primary resonances distant. Notice that because this is a
projection, trajectories can appear to cross. ( b ) A projection of a trajectory in a Liouvillian map
shows resonance-induced dispersion. The diagonal dashed line indicates the location of the primary
resonance. Close to this line the trajectory oscillates wildly and jumps from one invariant torus
s
to another. The map used is x’ = x 0.001(sinz 2cosy), y’ = y O.OOl(1.5 sinx‘ 2 . 5 ~ 0 z),
z’ = z + ~ ( C O Sy’ + sin x’).
+
+
+
+
by which the trajectory jumps from invariant torus to invariant torus. To compare
with figure 6 (a), in figure 6 ( b ) we show an example of the same resonance-induced
dispersion behaviour occurring in a two-action Liouvillian map. As we increase the
coupling a, the resonances have increasing influence, and the length of time for which
trajectories are captured into resonance increases. In figure 7 ( a ) we show another
projection of a stroboscopic map, this time for a larger value of a than in figure 6 ( a ) ,
but with the other parameters remaining unchanged. Trajectories are now captured
by resonances for much longer times, and by this means they can rapidly cross the
separatrix between the vortices at the equator and disperse throughout practically the
whole region between the spheres. This is shown graphically in figure 8 (a), which
illustrates a slice between -0.01 < y < 0.01 through the same stroboscopic map as
figure 7 (a). The stroboscopic map slice shows the evolution of a single trajectory, and
comparing it with the distribution of the resonances shown in figure 5 (a), we observe
that the only regions as yet unvisited in figure 8 ( a ) are close to the poles, and the
centres of the vortices that lie inside the resonance 0/1 shell and are not crossed by
any primary resonance. Resonance-induced dispersion into these regions will occur
slowly through weak high-order resonances that lie densely between the primary
resonances. To compare with this global mixing behaviour in the two-vortex case, in
figures 7 ( b ) and 8 ( b ) we show that for four-vortex flow, resonance-induced dispersion
operates separately within the inner and outer vortices: there is no mixing across the
separatrix between them, since this separatrix is concentric with, and therefore not
crossed by, any of the resonance shells.
Looking at figure 7 side by side with the a = 0 Hamiltonian chaos of figure 4 shows
how different resonance-induced dispersion appears in comparison; the absence of
276
J. H . E. Cartwright, M . Feingold and 0. Piro
1.o
1.o
0.5
0.5
z o
0
-0.5
-0.5
0
0.5
x
1.0
-1
n
1.o
0.5
0
X
6
FIGURE7. Projections of stroboscopic maps onto the plane y = r sine"sin = 0 for Re = 0.1 and
Other parameters are as in figure 5 (a), and a single trajectory is depicted in this
two-vortex flow. ( b ) Other parameters are as in figure 5 (b),and two trajectories are shown. This is
a four-vortex flow; there is no mixing across the separatrix between the inner and outer vortices.
a = 1". ( a )
KAM tori shows that we are dealing with chaos that is not a small perturbation
of an integrable Hamiltonian system. Whilst the Hamiltonian chaos of figure 4 is
produced by long-period flow, T = 12000, the resonance-induced dispersion of figure
7 is already present when the flow period is much shorter; T = 4. At these short
periodicities, the measure of Hamiltonian chaos is minute. In the case where p1 = p2
there is no Hamiltonian chaos when a = 0. Recall from our discussion of (3.19)-(3.21)
that when pl = p2 the coupling between ( r , @ and is weaker than otherwise. This
implies that when a # 0 and p1 = p2 all the resonances will have less influence than
normal. Furthermore, there is additional cancellation that takes place for r = roll
when pl = p 2 ; in this case hl(r) = h2(r) = 0 and the order-a terms cancel completely,
so that the 0/1 resonance is not present when p1 = p2. This anomalous behaviour
when pl = p 2 is why we have chosen throughout this paper to use examples where
p1# p2 for illustrating resonance-induced dispersion.
There are several parameters one can alter to control the rate of resonance-induced
dispersion. Tuning the resonances with (3.31) is one option. The stroboscopic map
projections of figures 6 ( a ) and 7 ( a ) were made with p1 = 0 and p2 = -8 so that the
0/1 primary resonance passed right through the centre of the region u] < r < 1 and
cut almost all invariant tori of (3.29), to ensure that resonance-induced dispersion
with a primary resonance would occur every time the trajectory wound once around
the vortex. Increasing the axis-switching period T would speed up the dispersion by
increasing the density of primary resonances, as we showed in figure 5 (b). If, on the
other hand, we choose to tune the system by changing ,iiand T so that all the primary
resonances lie outside the interval u] < r < 1, we would still have resonance-induced
dispersion, but operating only with the weaker secondary and higher resonances, so
that the dispersion would be slower with smaller jumps. Following the discussion
6
277
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
1
1
z o
0
-1
-1
1
0
X
1
0
X
1
FIGURE
8. Slices of stroboscopic maps between -0.01 < y < 0.01. ( a ) The parameter values and
initial conditions are as in figure 7 ( a ) ; the resonances are distributed as in figure 5 (a). Forty
thousand points (corresponding to many times this number of periods) are shown, all from the
same initial condition. This slice illustrates how effective resonance-induced dispersion is at mixing;
the only regions where the trajectory has not yet ventured are close to the poles, and the centres
of the vortices, which lie inside the 0/1 resonance. In time, the trajectory will disperse into these
regions too, through the action of the higher-order resonances that are dense throughout the space.
If, by altering p, the 0/1 resonance were tuned to pass right through the centres of the vortices, these
regions would be visited as frequently as others. ( b ) Two initial conditions are used to illustrate
dispersion in the trajectory with parameter values as in 7 ( b ) ;the resonances are distributed as in
figure 5 (b).The space is divided into two regions by the separatrix between the two vortices in each
hemisphere. There in no mixing between these two regions, but effective mixing occurs separately
within each region.
above, another possibility is to make p1 the same as or different to p2, as the
resonances are stronger when p1 # p2. We can also alter the axis separation a, since
the resonances gain strength with increasing CI,as the comparison of figure 6 ( a) with
figure 7 ( a ) shows.
For a different view of resonance-induced dispersion, we can look at the time
evolution of the stream function (3.29). If the adiabatic approximation of (3.26)-(3.27)
were exact, this would be constant in time. Instead, if resonance-induced dispersion
were occurring, we would expect to see that this graph should be almost flat when the
trajectory evolved on a level surface of an invariant torus, but these flat regions would
be interspersed with oscillations leading to sudden jumps, as each time the trajectory
approached a resonance intersecting the invariant torus it would be transported to
another level surface with a different value of H, rather like playing a game of snakes
and ladders. Figure 9 (a), for the same parameter values and initial conditions as
figure 6 (a), is such a graph, and indeed we observe this snakes-and-ladders behaviour
exactly as we have just described - we are seeing resonance-induced dispersion in
action. As an example to compare with figure 9 (a),we show in figure 9 ( b ) a similar
graph of resonance-induced dispersion in a two-action Liouvillian map.
We find numerically that the small-axis-separation approximation for biaxial unsteady spherical Couette flow remains good as long as a is less than around lo".The
calculations of invariant tori and resonant surfaces that we made for the small-axisseparation flow become increasingly poor approximations when the axis separation
278
J. H. E. Cartwright, M . Feingold and 0. Piro
,
-3,
I
-5
-6
0
4000
2000
Periods
-3.3
I
(b)
H - 3.4
-
-3.5
'
0
?",I
I
20 000
10 000
Iterations
FIGURE9. (a) Resonance-induced dispersion is illustrated here with a graph of the adiabatic
invariant (3.29) against time of the trajectory in figure 6 (a). ( b ) Resonance-induced dispersion
in a Liouvillian map is shown using a graph of the adiabatic invariant (2.7) against time of the
trajectory in figure 6 (b). In both (a) and ( b ) the adiabatic invariant remains nearly constant almost
all the time, but jumps chaotically from one invariant torus to another at each intersection with
a primary resonance. The general downward trend in the graphs corresponds to the trajectory
winding towards a vortex centre, as is visible in figures 6(a) and 6(b).
is made larger. For a greater than lo, the noise arising from the fact that the
small-axis-separation approximation is just an approximation starts to overwhelm the
signal. However, we can recuperate it by applying a running average to the noisy
signal from (3.29), as we show in figure 10. For tl more than about lo", this averaging
starts to lose its effectiveness, and consequently we are unable perform much analysis
for larger axis separations. We believe that the large-axis-separation flow is still of
two-action type: the interaction between 8 and will produce one action and one
angle variable, r being the other action. However since the large-axis-separation flow
ct
6
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
279
~ x 1 0 50
-6 I
0
I
2000
I
4000
Periods
-6
I
0
I
I
2000
4000
Periods
FIGURE10. Averaging at work; the initial conditions and parameter values here are as in figure
6 (a), except for the axis separation angle a, which is now lo". (a) The noisy adiabatic invariant
H (3.29) against time. The noise arises because H is only approximately the invariant, and the
approximation worsens with increasing a. ( b ) After performing a twenty-point running average on
the trajectory. The running average has smoothed out the noise substantially, so that we can now
see the jumps much more clearly. This shows that even in cases where we do not possess the exact
adiabatic invariant, it is still possible using averaging to see the characteristic two-action jump
behaviour at work.
+
is not a small perturbation of a (2 1)-dimensional flow, we are unable to obtain
analytical expressions for the action and angle variables in the large-axis-separation
case in the same way as was possible for small-axis-separation flow.
4. Chaotic advection in three-dimensional unsteady flows
Following our classification in $2 of three-dimensional unsteady flows by the
number of fast angle and slow action variables, biaxial unsteady spherical Couette
280
J. H. E. Cartwright, M. Feingold and 0. Piro
flow is of the two-action class. Three-dimensional flows often arise from what we
have termed (2 + 1)-dimensional flows by the coupling of a one-dimensional primary
flow to a two-dimensional secondary flow, giving one variable of one type and two
variables of the other. Generically this will preclude zero-action and three-action
flows, but a large class of three-dimensional unsteady flows are two-action flows,
and one-action flows will be equally common (Mezib & Wiggins 1994). With biaxial
unsteady spherical Couette flow the relationship between the primary and secondary
flow rates depends on the Reynolds number - the secondary flow is hundreds to
thousands of times slower than the primary flow when Re = 0.1 - so that although
it would be possible in our model to increase the Reynolds number sufficiently to
give two fast secondary variables and one slow primary variable, making the flow of
one-action type, doing so would make the low-Reynolds-number approximation we
started with invalid, landing us squarely in the turbulent regime (Buhler 1992), Other
possible candidate flows for obtaining a one-action three-dimensional unsteady flow
model that can both be treated analytically and is experimentally realizable should
be sought out and investigated.
We have seen in biaxial unsteady spherical Couette flow that the zero-axisseparation limit of a = 0 can lead us either to two-dimensional steady flow without
chaos, if p1 = p2, or to two-dimensional unsteady flow with chaos, if p1 # p2. The
limit we cannot explore with biaxial unsteady spherical Couette flow corresponds to
the transition between three-dimensional unsteady flow and three-dimensional steady
flow. One way to arrive at a three-dimensional steady flow model using spherical
Couette flow would be to displace the spheres so that they would no longer be coaxial,
like the cylinders in journal-bearing flow. In this way we would break the &symmetry
and obtain a three-dimensional steady flow, spherical-bearing flow, which could then
be modulated in time using different p1 and p2 to give a three-dimensional unsteady
flow. The only work we know of on this geometry is that of Wannier (1950), who
however only treats the the problem of spherical-bearing flow in the limit of a very
small gap between the spheres, such that the radial component of the motion can
be assumed to be zero. This limit of the Stokes equation leads to the Reynolds
equations of lubrication theory, which give a two-dimensional flow in 0 and 4. Munson (1974) solves perturbatively the case where the spheres are displaced along their
common axis, so that they are eccentric but remain coaxial. This however retains the
&symmetry, so it is another example of a (2 1)-dimensional steady flow rather than
a three-dimensional steady flow.
Another possibility for investigating the transition between three-dimensional
steady and unsteady flow is given by the flow introduced by Bajer & Moffatt
(1992), who treat the three-dimensional steady flow between two concentric spheres
rotating about different axes. This flow is particularly interesting because it exhibits
the consequences of the breakdown of the Hamiltonian description of steady threedimensional flows that occurs at stagnation points of the flow. This phenomenon
is similar in origin to the resonance-induced dispersion we have been investigating
here in three-dimensional unsteady flows. In both cases averaging fails in certain
regions of space. In three-dimensional unsteady flows these regions are generically two-dimensional resonant surfaces, whereas in three-dimensional steady flows
they are zero-dimensional stagnation points. However, despite the less important
r81e, occasioned by their lower dimensionality, that these regions play in threedimensional steady flow, in appropriate circumstances the global topology of the
flow can be altered by the breakdown of averaging at the stagnation points, leading
to what Bajer & Moffatt have termed trans-adiabatic drift (Bajer & Moffatt 1990;
+
Chaotic advection in three-dimensional unsteady incompressible laminar $ow
28 1
Bajer 1994). Trans-adiabatic drift can be considered to be the equivalent in threedimensional steady flows of resonance-induced dispersion. It should thus be particularly interesting to model the addition of time dependence to the flow described
above. We should point out that adiabatic descriptions of processes that are valid
everywhere except in certain neighbourhoods are ubiquitous in many areas of physics,
and lead to graphs of the type of figure 9 being seen in many contexts.
Occupying as they do a position somewhere between symplectic maps of two and
four dimensions from two- and three-degree-of-freedom Hamiltonian systems, the
Liouvillian maps that underlie three-dimensional unsteady flows can exhibit types
of behaviour analogous to both their lower- and higher-dimensional siblings. An
important difference between Hamiltonians of two and three degrees of freedom is
that Arnold diffusionf exists in the latter but not in the former. With two degrees
of freedom, the invariant KAM tori completely separate different regions of phase
space. This is no longer true when another degree of freedom is added, and as
a result Arnold diffusion can take place throughout the phase space. This occurs,
albeit extremely slowly, for arbitrarily small values of the nonlinearity parameter E .
Resonance-induced dispersion in two-action Liouvillian maps, although reminiscent of
Arnold diffusion in the four-dimensional volume-preserving maps from three-degreeof-freedom Hamiltonians, is engendered by a different mechanism. Whilst Arnold
diffusion occurs through an interconnected web of resonances, in resonance-induced
dispersion the available space is covered using the invariant tori, and the resonant
surfaces are a means to jump from one torus to another. The consequence of this
difference is that the rates of the two processes are radically different: with nonlinearity
E , an estimate of the upper bound on the rate of Arnold diffusion is O(exp(--E-'/2)),
whilst previous work on Liouvillian maps shows that resonance-induced dispersion
has an O(c2)dependence (Piro & Feingold 1988).
The presence of resonance-induced dispersion in two-action-type flows, and its
absence in those of one-action type, is from the mixing and transport point of view
the crucial difference between these two classes of flows. In the one-action class
of three-dimensional unsteady flows, as with two-dimensional unsteady flows, the
existence of KAM-type theorems proves that complete mixing will not generically be
achievable, as the invariant tori present barriers to transport. On the other hand, the
addition of a third dimension to a two-dimensional unsteady flow leads in the twoaction case to much faster dispersion than with two-dimensional flow alone, owing
to the presence of resonance-induced dispersion. This should lead to better mixing.
What we mean by efficient mixing here is that the density of an initial concentration
of passive scalars should rapidly tend to a constant value throughout the region being
mixed. This property is related to the Lyapunov exponents of the flow, one of which
must be positive to allow the exponential separation of neighbouring trajectories
that is necessary for chaos. We shall be reporting in future work investigations of
Lyapunov exponents and mixing efficiencies in biaxial unsteady spherical Couette
flow and other three-dimensional unsteady flows. Even a small coupling between
primary and secondary flows is effective in two-action flows in producing a large
improvement in dispersion over two-dimensional chaotic advection alone through the
action of resonance-induced dispersion, particularly when the flow period is relatively
t One of the terminological differences between nonlinear dynamics and fluid dynamics is that
diffusion in nonlinear dynamics refers to a process that in fluid dynamics is called dispersion
(diffusion in fluid dynamics carrying connotations of molecular diffusion). In previous papers (Piro
& Feingold 1988; Feingold et al. 1989; Cartwright et al. 1994b, 1995) we used the term diffusion in
the dynamical systems sense.
282
J. H. E. Cartwright, M . Feingold and 0. Piro
short when there is very little chaotic advection in two dimensions. In practical terms,
we have seen that even a wobble from a bad bearing between the two spheres leading
to a free play of 0.1" is enough to produce a spectacular growth in dispersion rate.
Although we have run some of our plots for long times, by adjusting the parameters
it is possible to arrive at a flow in which the effects of resonance-induced dispersion
are visible after only a handful of periods.
Biaxial unsteady spherical Couette flow forms a bridge between theory and experiment for investigating chaotic advection in three-dimensional unsteady flows : we
have an analytical solution for the stream function and the flow can be realized in
the laboratory. Journal-bearing flow has performed this same function extremely well
for two-dimensional unsteady flows. We hope that the model we have presented
might spur similar interest in the analysis and investigation of chaotic advection in
three-dimensional unsteady flows, and that the challenge of devising suitable apparatus for the experimental observation of resonance-induced dispersion and the other
phenomena described here will be taken up.
We should like to thank Konrad Bajer and an anonymous referee for their useful
comments and suggestions about the first draft of this paper. M.F. would like to
thank the physics department of the Universitat de les Illes Balears for hospitality
during a month-long visit in which the idea for this paper was born. O.P. and J.H.E.C.
acknowledge the support of the Spanish Direccih General de Investigacibn Cientifica
y Tkcnica, contract numbers PB94-1167 and PB94-1172 and European Union Human
Capital and Mobility contract number ERBCHBICT920200. M.F. acknowledges the
support of the Israel Science Foundation administered by the Israel Academy of
Sciences and Humanities.
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